Wanderlust World Wanderlust World In quantum mechanics, a number state (or Fock state) is a precise quantum state of a system with a definite number of identical particles—or quanta of energy—in a given mode. It's the quantum analog of "exactly n photons in this cavity" or "exactly n phonons in this lattice vibration."
In a Hilbert space spanned by basis vectors |n⟩ (n = 0, 1, 2...), a number state is an eigenstate of the number operator N̂ = a†a, where a† and a are creation and annihilation operators:
N̂|n⟩ = n|n⟩
The probability density is |⟨x|n⟩|². Number states are orthogonal: ⟨m|n⟩ = δₘₙ.
The textbook case. Energy eigenvalues: Eₙ = ħω(n + ½). The ground state |0⟩ has zero-point energy. Higher n = more energy quanta. Wavefunctions are Hermite polynomials × Gaussian.
Each field mode has its own number states. |nₖ⟩ means n particles with momentum k. The vacuum is |0⟩ for all modes. Multi-particle states are tensor products: |n₁, n₂, ...⟩.
Electron states in atoms are labeled by quantum numbers (n, l, mₗ, mₛ)—not single number states, but the principal quantum number n shares the "counting" concept.
Number states provide a discrete, countable basis for continuous quantum fields. They're essential for quantum optics (photon counting), quantum information (qubit encodings), condensed matter (phonons, magnons), and particle physics. Coherent states (laser light) are superpositions of number states—understanding the latter unlocks the former.